Question

What happens to the discretization and the roundoff errors as the step size is decreased?

Short answer

Answer: Decreasing the step size reduces discretization error by providing better approximations of the continuous function and increasing the number of points representing it. However, smaller step sizes also result in more calculations, which can accumulate roundoff errors and potentially lead to larger overall errors. The optimal step size balances these two types of errors to produce the most accurate numerical solution possible.

Step-by-step solution

Understanding Discretization Error

Discretization error is the error introduced when we approximate a continuous function or process by a discrete set of points or intervals. For example, when solving a differential equation using a numerical method (like Euler's method), we divide the interval of interest into smaller steps and approximate the continuous function at discrete values. Discretization error is directly related to the step size, as smaller step sizes can lead to better approximations of the continuous function.

Understanding Roundoff Error

Roundoff error is the error caused by representing real numbers in a finite number of digits, which is necessary when performing calculations on a computer. When performing arithmetic operations (addition, subtraction, multiplication, and division), the roundoff error can accumulate, especially if the number of calculations is large. Smaller step sizes result in more calculations, which might cause roundoff errors to become significant.

Analyzing the effect of decreasing step size on discretization error

As the step size decreases, the approximation of the continuous function becomes better, since more points are used to represent it. Consequently, the discretization error will decrease with decreasing step size. In general, the relationship between the discretization error and step size can be modeled as: \[E_{discretization} \propto h^p\] where \(E_{discretization}\) is the discretization error, \(h\) is the step size, and \(p\) is the order of the numerical method (e.g., \(p = 1\) for Euler's method).

Analyzing the effect of decreasing step size on roundoff error

As we decrease the step size, the number of calculations required to represent the continuous function increases. As a result, the roundoff errors in each calculation can accumulate and potentially result in a larger overall error. The relationship between roundoff error and step size can be modeled as: \[E_{roundoff} \propto \frac{1}{h}\] where \(E_{roundoff}\) is the roundoff error and \(h\) is the step size.

Balancing Discretization and Roundoff Errors

When choosing an appropriate step size for a numerical method, it is essential to balance the effects of discretization error and roundoff error. Decreasing the step size will reduce the discretization error but may increase roundoff error. If the step size is too small, roundoff errors become dominant, and the accuracy of the numerics deteriorates. Conversely, if the step size is too large, discretization errors become dominant. In summary, as the step size decreases, discretization error decreases, and roundoff error generally increases. The optimal step size balances these two types of errors to produce the most accurate numerical solution possible.